Antisymmetry

Antisymmetry, second quantization and things related to it are something I kind of stayed away so far. And now that I have finally taken a look at it, I notice how nice the math behind it actually is. And the important things that many things work in general and do not require orbitals.

We consider functions of the following form where A is some set and N a natural number[1]:

Let V be the vector space of all such functions. And let's look at operators [2] acting on V. First we could consider the Transposition operator

This is a well defined mapping and the application is straight forward, e.g.

And if you think about it some more you notice that it is linear.

And if I am correct it is both Hermitian and orthogonal with the usual skalar product. This would mean that all the eigenvalues are either 1 or -1. Either way, we are interested in the eigenspaces corresponding to the eigenvalue -1, i.e. functions that are antisymmetric with respect to the transposition. The intersection of all these eigenspaces V_A is the set of functions that are antisymmetric with respect to all the transpositions. It is a vector space since it is formed as a intersection of vector spaces.

All fermion wave functions that comply with the Pauli principle have to be taken out of this space.

Next it helps to be a little bit more general and to introduce the permuation operator (related to a permuation σ out of the symmetric group S_N) as a generalization of the transposition operator

This operator is also linear and I think also Hermitian and orthogonal.

Then you could write the antisymmetric space also like this

since every transposition is a permutation with negative sign and every permutation can be formed out of transpositions and the sign corresponds to how many you take.

Well let's define one more operator, the antisymmetrizer

You can show that this operator is a projection operator. This is equivalent to the condition that applying it twice is the same as applying it once, since the vector is already projected after the first time. The proof is also straight forward. Applying it twice leads to a double sum

But this just means that you sum n! times over all permutations, and it turns out

And next you could prove that the space it projects into, is just the V_A we had before. You have to show that any projected function fullfils the condition for V_A above and that every function which fullfills the condition remains unchanged which is both quite straight forward.

Maybe I should also add why these results are nice. I did have the expression for the Slater determinant before. But by considering that this is nothing but a (normalized) projection of an orbital product into the antisymmetric space it is easier to deal with it. It is no longer some weird expression but nothing but a linear operator. And things like adding Slater determinants are much clearer when you consider this. And that is why it is cool.

And actually as a next step you can consider commutation. These operators commute with the Hamiltonian because they are nothing but relabelling of equivalently treated electrons. Only for that reason you know that eingenfunctions of the Hamiltonian which are also antisymmetric even exist. You restrict the Hamiltonian eigenfunctions to the ones with a -1 eigenvalue of the antisymmetrizer. In the next step you may also do some restrictions according to spatial symmetry - the interacting spaces will transform like irreducible representations of the symmetry group. And finally spin. In the spin case the interacting spaces transform like representations of the unitary group (and I am kind of trying to understand why that is).

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[1] Physically spoken N is the number of particles and A is the set of possible coordinates of the particle.

[2] As I probably said before: An operator is a function that makes a function out of a function. Or if you don't like the word "function", it is a mapping between two vector spaces.

Hubble

For some reason CNN is one of the only channels that I can access from my room here in Prague that is not either in czech or featuring arabic telephone sex commercials. One thing I saw there were some pretty amazing bulls and bears that reminded me of N. N. Taleb. But I also saw some exciting space shuttle features. I like space shuttles but I am wondering what they are actually good for. (I am not an expert on space shuttles but I did once hear an astronomer talk about them.)

As I understand it, a space shuttle is like a Porsche. It's exciting, it's prestigious, but it does not really get you anywhere you could not go without it. (The advantage is that a space shuttle is not as noisy and does not cause traffic jams that I have to carefully bypass on my bicycle without collecting any mirrors.) The question that the people at CNN never asked any of the space shuttle experts is how sending up a 7 austronauts and assembling a telescope in deep space compares to assembling the telescope down here and sending it up by itself. I guess a space shuttle is a nice piece of science fiction, without the need of fiction, but probably not the most cost efficient way of having a telescope in the sky.

CERN

I should not critize my home country but this was a masterpiece: We are apparently quitting CERN! Especially now that they build the LHC.

Well the upside is: When the black hole is created that swallows the whole world, we can say it was not our fault.

Edit: and the second upside is that they will have about 20 million extra euros that they can put into my stipend.

Edit2: We are not leaving CERN thanks to public pressure.

DNA

Ever since Jurassic Park we know that DNA looks pretty cool. Here is a piece of B-DNA.[1]

Or with thicker sticks:


In fact there is of course not so much space between the bases. Here is what it looks like when you draw the van der Waals spheres for the bases. It is actually quite difficult to model this stacking interaction because it is almost pure dispersion. From a computational point of view, dispersion is electron correlation. If the effect you are interested in is pure electron correlation, then you'd better model it really well. Hartree-Fock gives you zero dispersion. MP2 is great if you kind of want to include some electron correlation but it is not accurate enough in this case. What they are actually doing is high level coupled cluster CCSD(T) extrapolated to the complete basis set limit.


Another interesting question is how defects are propagated in such a framework: holes, electrons, or electron hole pairs. First of course you want to preserve the integrity of your genome. And second you want to make nano-robots. DNA is already a self-assembling structure with molecular recognition. If charge transport is better understood and conductive analogues are found, DNA will kick nanotubes' ass.

By the way: for printing out pymol graphics (see again [1]), the ray_trace_mode setting is nice. Especially:

set ray_trace_mode=1

Then you get some nice black frames instead of fuzzi ends in the print.

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[1] from the crystal structure 1BNA. And drawn with pymol. Maybe I should eventually switch from pymol because they are becoming more and more commercial. But then I guess they would not mind me as a little personal user.

Check every day

I have three more items that you should put on your "Check every day before you start working list":

• Abstruse Goose
• Cyanide and Happiness
• Saturday Morning Breakfast Cereal

What should already be on there:

• PhD comics
• xkcd
• dinosaur comics
  

exp(K)

It is time for some nice extra math, with some proofs and applications. I have two theorems that don't look obvious at first. And initially I kind of tried to avoid them.

The first one is that the matrix exponential of a real antisymmetric matrix is an orthogonal matrix with determinant +1

The second one is that for any matrix K (for example the anti-symmetric one from I) and a matrix A the following holds.

Or explicitely:

This is an expansion in the powers of K. And if K is small, you can cut it at some point. The reason why this is useful is that it gives you a direct possibility to manipulate orthogonal matrices. You don't have to worry about making the matrix orthogonal but you just choose any paramters for K and exp(K) will be orthogonal. II gives the possibility to carry out the basis transformation in an efficient way.

The first part of I is actually very straight forward if you write it down like this. It is provided that the limit of a transposed sequence is the transposition of the original limit and that the inverse is formed as shown.

I will prove the second part with the spectral theorem (maybe you can also do this in a more direct way). If we move into the complex space. K is now correctly spoken anti-Hermitian, and we know that a unitary matrix U exists which diagonalizes K. The eigenvalues of K are purely imaginary (yet quite important).

The exponential is given according to

At this point you can also notice that exp(K) is unitary because it is formed as a product of three unitary matrices.

Can we say anything about the determinant? Yes of course.

The determinants of the mutually inverse matrices UH and U cancel out. The determinant of a diagonal matrix is just the product of the diagonal terms.

The sum of the eigenvalues is the trace of K. In a real antisymmetric matrix all diagonal elements have to be zero (therefore also the trace).

And that is what we wanted to see.

The proof shows that exp(.) is a mapping between the set of real antisymmetric matrices and the set of real orthogonal matrices with determinant +1 (and of course that -exp(.) is the mapping to matrices with det=-1). In fact it would also be necessary to show that this mapping is bijective (i.e. that there is a 1:1 correspondence). This is apparently true but I don't think the proof is so simple. Interestingly it does not work with complex anti-Hermitian matrices. In this case there are n extra purely imaginary values in the diagonal. And the mapping is no longer injective for example:

Point I was the elegant part. Point II is more of the index juggling variety but a nice piece of complete induction. I will probably put that up soon, too.